dblogr/ Publications Academic agData Blog

Calculating Heritability

An R tutorial on how to calculate heritability

UNDER CONSTRUCTION!!!!

Introduction

# devtools::install_github("derekmichaelwright/agData")
library(agData)
library(lme4)
library(ggridges)

Formulas

Phenotypic variance, the variation for a particular trait, can be attributed to both genetic and environmental factors. This leads to the concept of Heritability, the degree of variation in a phenotypic trait in a population that is due to genetic variation between individuals in that population. There are two types of heritability used by researchers: Broad Sense Heritability (\(H^2\)) and Narrow Sense Heritability (\(h^2\)).

\(V_P=V_G+V_E\)

where:

  • \(V_P\) = Phenotypic variance
  • \(V_G\) = Genetic variance
  • \(V_E\) = Environmental variance

Broad Sense Heritability

\(H^2=\frac{V_G}{V_P}=\frac{V_G}{V_G+V_E}\)

Narrow Sense Heritability

\(h^2=\frac{V_A}{V_P}\)

where:

  • \(V_A\) = Additive variance

Breeders Equation:

\(R=h^2S\)

where:

  • \(S\) = Selection differential
  • \(R\) = Response to selection

Tomato Example

TBRTQuality.csv

# Prep data
myCaption <- expression(paste("derekmichaelwright.github.io/dblogr/ | Data: Collins ", 
                               italic("et al"), ". (1930)"))
#
dd <- read.csv("TBRTQuality.csv") %>% 
  mutate(Rep = factor(Rep),
         Expt = paste(Loc, Year))

Brix: a measure of the dissolved solids in a liquid, and is commonly used to measure dissolved sugar content of an aqueous solution. One degree Brix is 1 gram of sucrose in 100 grams of solution and represents the strength of the solution as percentage by mass.

# Plot
mp <- ggplot(dd, aes(x = Rep, y = Brix, color = Rep)) +
  geom_quasirandom(alpha = 0.7) +
  facet_grid(. ~ paste(Year, Loc)) +
  scale_color_manual(values = c("darkgreen","steelblue")) +
  theme_agData(legend.position = "none") +
  labs(caption = myCaption)
ggsave("heritability_1_01.png", mp, width = 6, height = 4)

# Prep data
myColors <- c("darkgreen", "steelblue", "darkred", "darkblue")
xx <- dd %>% group_by(Line, Expt) %>% 
  summarise(Mean = mean(Brix, na.rm = T),
            Max  =  max(Brix, na.rm = T),
            Min  =  min(Brix, na.rm = T)) %>% 
  ungroup() %>% arrange(Mean) %>%
  mutate(Line = factor(Line, levels = unique(.$Line))) %>%
  filter(!is.na(Mean))
# Plot
mp <- ggplot(xx, aes(x = Line, color = Expt)) +
  geom_point(aes(y = Mean)) +
  geom_segment(aes(xend = Line, y = Min, yend = Max)) +
  scale_color_manual(values = myColors) +
  facet_grid(Expt ~ .) +
  theme_agData(legend.position = "bottom",
               axis.text.x = element_text(angle = 45, hjust = 1, size = 4)) +
  labs(y = "Brix", caption = myCaption)
ggsave("heritability_1_02.png", mp, width = 10, height = 6)

# Prep data
x1 <- xx %>% filter(Expt == "CA 2010") %>%
  top_n(-Mean, n = 10)
x2 <- xx %>% filter(Expt == "CA 2010") %>%
  top_n(Mean, n = 10)
yy <- dd %>% filter(Line %in% c(x1$Line, x2$Line)) %>%
  mutate(Line = factor(Line, levels = c(x1$Line, x2$Line)),
         Group = ifelse(Line %in% x1$Line, "Low", "High"),
         Group = factor(Group, levels = c("Low", "High")))
# Plot
mp <- ggplot(yy, aes(x = Group, y = Brix, color = Group)) +
  geom_quasirandom(alpha = 0.8) +
  facet_grid(. ~ Expt) +
  scale_color_manual(values = c("darkred", "steelblue")) +
  theme_agData(legend.position = "none",
               axis.text.x = element_text(angle = 45, hjust = 1)) +
  labs(x = NULL, caption = myCaption)
ggsave("heritability_1_03.png", mp, width = 6, height = 4)

Lets run an anova to determine environmental variance.

# ANOVA
myfit <- lmer(Brix ~ (1 | Line) + (1 | Loc) + (1 | Year) + (1 | Line:Loc) + (1 | Line:Year), data = dd)
myfit_aov <- summary(myfit)
myfit_aov
## Linear mixed model fit by REML ['lmerMod']
## Formula: Brix ~ (1 | Line) + (1 | Loc) + (1 | Year) + (1 | Line:Loc) +      (1 | Line:Year)
##    Data: dd
## 
## REML criterion at convergence: 1742.1
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.0258 -0.6133 -0.0533  0.5293  5.0653 
## 
## Random effects:
##  Groups    Name        Variance Std.Dev.
##  Line:Year (Intercept) 0.010211 0.10105 
##  Line:Loc  (Intercept) 0.009482 0.09738 
##  Line      (Intercept) 0.191931 0.43810 
##  Year      (Intercept) 0.031639 0.17787 
##  Loc       (Intercept) 0.188458 0.43412 
##  Residual              0.257820 0.50776 
## Number of obs: 967, groups:  Line:Year, 284; Line:Loc, 281; Line, 143; Year, 2; Loc, 2
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)   5.2817     0.3343    15.8

\(Heritability = \frac{var(Line)}{ var(Line) + \frac{var(Line:Loc)}{2} + \frac{var(Line:Year)}{2} + \frac{var(Residual)}{4}}\)

\(H^2 = \frac{0.1919}{0.1919 + \frac{0.0095}{2} + \frac{0.0102}{2} + \frac{0.2578}{4}} = 0.72\)

Vg <- myfit_aov$varcor$Line[1]
Ve <- myfit_aov$varcor$`Line:Loc`[1] / 2 + 
      myfit_aov$varcor$`Line:Year`[1] / 2 + 
      myfit_aov$sigma^2 / 4
Vg / (Vg + Ve)
## [1] 0.7209154
myBLUPs <- ranef(myfit)$Line
ggplot(myBLUPs, aes(x = `(Intercept)`)) +
  geom_histogram(color = "black", fill = "darkred", alpha = 0.7)

Tobacco Flowers

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1193657/pdf/164.pdf

https://www.youtube.com/watch?v=NZ3V7V-yJbw

dd <- read.csv("east1916.csv") %>%
  mutate(Name = factor(Name, levels = rev(unique(.$Name)))) %>%
  gather(Size, Num, 7:ncol(.)) %>%
  mutate(Size = as.numeric(gsub("mm", "", Size))) %>%
  filter(!is.na(Num))

xx <- dd %>% filter(Selection != "Large")
mp <- ggplot(xx, aes(x = Size, y = Name, fill = Num, label = Num)) +
  geom_tile(color = "black", alpha = 0.7, width = 3) +
  geom_text() +
  facet_grid(Year ~ ., scale = "free_y", space = "free_y") +
  scale_fill_gradient(low = "grey", high = "darkred") +
  theme_agData(legend.position = "none") +
  labs(caption = myCaption)
ggsave("heritability_2_01.png", mp, width = 10, height = 6)

xx <- dd %>% filter(Selection != "Small")
mp <- ggplot(xx, aes(x = Size, y = Name, fill = Num, label = Num)) +
  geom_tile(color = "black", alpha = 0.7, width = 3) +
  geom_text() +
  facet_grid(Year ~ ., scale = "free_y", space = "free_y") +
  scale_fill_gradient(low = "grey", high = "darkred") +
  theme_agData(legend.position = "none") +
  labs(caption = myCaption)
ggsave("heritability_2_02.png", mp, width = 10, height = 6)

# Prep data
myLines <- c("No. 383", "No. 330", "No. 383 & No. 330", "No. 383 x 330", "No. (383 x 330) 1" ,
             "No. (383 x 330) 1-3",  "No. (383 x 330) 1-3-1", "No. (383 x 330) 1-3-1-1")
xx <- dd %>% filter(Selection != "Large", Name %in% myLines) %>%
  group_by(Name, Size) %>% summarise(Num = sum(Num)) 
# Plot
mp <- ggplot(xx, aes(x = Size, y = Name)) +
  geom_density_ridges() +
  theme_agData(legend.position = "none") +
  labs(caption = myCaption)
ggsave("heritability_2_03.png", mp, width = 10, height = 6)
xx <- dd %>% 
  group_by(Name, Selection, ParentPop, ParentSize) %>%
  summarise(Mean = sum(Num * Size) / sum(Num),
            Mean = round(Mean, 1)) %>%
  ungroup() %>%
  left_join(select(., ParentPop=Name, ParentMean=Mean), by = "ParentPop") %>%
  mutate(R = abs(Mean - ParentMean),
         S = abs(ParentSize - ParentMean),
         h2 = R / S,
         h2 = round(h2, 2))
DT::datatable(xx)

By rearranging the breeders formula, narrow sense heritability can be determined by calculating the slope of a regression line of progeny mean as a function of parental mean.

\(h^2=R/S\)

myfit <- lm(Mean ~ ParentSize, xx)
mp <- ggplot(xx %>% filter(Selection != ""), 
             aes(x = ParentSize, y = Mean, color = Selection)) +
  geom_smooth(method = "lm", se = F, color = "black", alpha = 0.7) +
  geom_point(size = 2.5, alpha = 0.7) +
  scale_color_manual(values = c("darkgreen","steelblue")) +
  ylim(c(min(xx$Mean), max(xx$Mean))) + 
  xlim(c(min(xx$Mean), max(xx$Mean))) +
  theme_agData(legend.position = "bottom") +
  labs(title = paste("h^2 =", round(myfit$coefficients[2], 2)),
       x = "Parental Mean", y = "Progeny Mean", caption = myCaption)
ggsave("heritability_2_04.png", mp, width = 6, height = 4)

myfit_L <- lm(Mean ~ ParentSize, xx %>% filter(Selection == "Large"))
myfit_S <- lm(Mean ~ ParentSize, xx %>% filter(Selection == "Small"))
mp <- ggplot(xx %>% filter(Selection != ""), 
             aes(x = ParentSize, y = Mean, color = Selection)) +
  geom_smooth(method = "lm", se = F, color = "black", alpha = 0.7) +
  geom_point(size = 2, alpha = 0.7) +
  facet_grid(. ~ Selection) +
  scale_color_manual(values = c("darkgreen","steelblue")) +
  ylim(c(min(xx$Mean), max(xx$Mean))) + 
  xlim(c(min(xx$Mean), max(xx$Mean))) +
  theme_agData(legend.position = "none") +
  labs(title = paste("Large h^2 =", round(myfit_L$coefficients[2], 2),
                    "| Small h^2 =", round(myfit_S$coefficients[2], 2)),
       x = "Parental Mean", y = "Progeny Mean", caption = myCaption)
ggsave("heritability_2_05.png", mp, width = 6, height = 4)

© Derek Michael Wright